Valerio Faraoni, Jeremy Coté
A generalization of the Hawking-Hayward quasilocal mass to scalar-tensor gravity is compared, in vacuo and for asymptotically flat stationary geometries, with a recent multipole expansion of the gravitational field. The quasilocal mass seen at spatial infinity coincides with the monopole term, lending credibility to this construct.
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The Bel-Robinson tensor $B_{αβμν}$ gives a positive definite gravitational energy in the quasilocal small sphere limit approximation. However, there is an alternative tensor $V_{αβμν}$ that was proposed recently that offers the same positivity as $B_{αβμν}$ does. We have found that $V_{αβμν}$ is the unique alternative tensor with $B_{αβμν}$ which implies that these two tensors are a basis for expressions that have the desirable non-negative gravitational energy in the small sphere limit. In other words, the `energy-momentum' density according to $B_{αβμν}$ and $V_{αβμν}$ are on equal footing a…
The Bel-Robinson tensor $B$ and the tensor $V$ have the same quasilocal energy-momentum in a small sphere. Using a pseudotensor approach to evaluate the energy-momentum in a half-cylinder, we find that $B$ and $V$ have different values, not proportional to the "Bel-Robinson energy-momentum". Furthermore, even if we arrange things so that we do get the same "Bel-Robinson energy-momentum" value, the angular momentum gives different values using $B$ and $V$ in a half cylinder. In addition, we find that $B$ and $V$ have a different number of independent components. The fully trace free property of…
Following an existing procedure in general relativity, the turnaround radius of a spherical structure is studied in scalar-tensor gravity using a new prescription for the analog of the Hawking-Hayward quasilocal mass in this class of theories.
The M$ø$ller tetrad gravitational energy-momentum expression was recently evaluated for a small vacuum region using orthonormal frames adapted to Riemann normal coordinates. However the result was not proportional to the Bel-Robinson tensor $B_{αβμν}$. Treating a modified quasilocal expressions in a similar way, we found one unique combination that gives a multiple of $B_{αβμν}$ which provides a non-negative gravitational energy-momentum in the small sphere approximation. Moreover, in addition to $B_{αβμν}$, we found a certain tensor $S_{αβμν}+K_{αβμν}$ which gives the same "energy-momentum" d…
A recent generalization of the Hawking-Hayward quasilocal energy to scalar-tensor gravity is adapted to general spherically symmetric geometries. It is then applied to several black hole and other spherical solutions of scalar-tensor and $f({\cal R}) $ gravity. The relations of this quasilocal energy with the Abreu-Nielsen-Visser gauge and the Kodama vector are discussed.
Finite-cutoff holography and quasilocal thermodynamics of BTZ black holes in a cavity
本文提出有限截断形式的BTZ黑洞热力学，将圆环腔视为真实反 holographic 屏幕，结合准局部引力系统和AdS_3中的有限半径holography，探讨静态和旋转BTZ几何的热力学。
A new generalization of the Hawking-Hayward quasilocal energy to scalar-tensor gravity is proposed without assuming symmetries, asymptotic flatness, or special spacetime metrics. The procedure followed is simple but powerful and consists of writing the scalar-tensor field equations as effective Einstein equations and then applying the standard definition of quasilocal mass.
For describing the non-negative gravitational energy-momentum in terms of a pure Bel-Robinson type energy-momentum in a quasilocal 2-surface, both the Bel-Robinson tensor $B$ and tensor $V$ are suitable. We found that this Bel-Robinson type energy-momentum can be modified such that it satisfies the Lorentz covariant, future pointing and non-spacelike properties. We find that these particular energy-momentum properties can be obtained from (i): $B$ or $V$ plus a tensor $S$ in a quasilocal small cube limit, or (ii): directly evaluating the energy-momentum of $B$ or $V$ in a quasilocal small box …
The Misner-Sharp-Hernandez mass defined in general relativity and in spherical symmetry has been recognized as having a Newtonian character in previous literature. In order to better understand this aspect we relax spherical symmetry and we study the generalization of the Misner-Sharp-Hernandez mass to general spacetimes, i.e., the Hawking quasilocal mass. The latter is decomposed into a matter and a pure Weyl contribution. The decomposition of the Weyl tensor into an electric part (which has a Newtonian counterpart) and a magnetic one (which does not) further splits the quasilocal mass into "…
Quasilocal definitions of stress-energy-momentum---that is, in the form of boundary densities (in lieu of local volume densities)---have proven generally very useful in formulating and applying conservation laws in general relativity. In this essay, we take a first basic look into applying these to cosmology, specifically using the Brown-York quasilocal stress-energy-momentum tensor for matter and gravity combined. We compute this tensor and present some simple results for a flat FLRW spacetime with a perfect fluid matter source.
The quasilocal energy of gravitational and matter fields in a spatially bounded region is obtained by employing a Hamilton-Jacobi analysis of the action functional. First, a surface stress-energy-momentum tensor is defined by the functional derivative of the action with respect to the three-metric on ${}^3B$, the history of the system's boundary. Energy density, momentum density, and spatial stress are defined by projecting the surface stress tensor normally and tangentially to a family of spacelike two-surfaces that foliate ${}^3B$. The integral of the energy density over such a two-surface $…
Multipole analysis on stationary massive vector and symmetric tensor fields
本文利用对称无迹形式获得了稳态有质矢量与对称张量场的多极展开及源多极矩封闭表达式，揭示了场参数的类汤川依赖性及附加多极矩的存在。
The field equations of $f(R)$ gravity are rewritten in the form of obvious wave equations with the stress-energy pseudotensor of the matter fields and the gravitational field as its source under the de Donder condition. The method of multipole analysis in terms of irreducible Cartesian tensors is applied to the linearized $f(R)$ gravity, and its multipole expansion is presented explicitly. In this expansion, the tensor part is symmetric and trace-free and is the same as that in General Relativity, and the scalar part predicts the appearance of monopole and dipole radiation in $f(R)$ gravity, a…
We show that in a generic scalar-tensor theory of gravity, the ``referenced'' quasilocal mass of a spatially bounded region in a classical solution is invariant under conformal transformations of the spacetime metric. We first extend the Brown-York quasilocal formalism to such theories to obtain the ``unreferenced'' quasilocal mass and prove it to be conformally invariant. The appropriate reference term in this case is defined by generalizing the Hawking-Horowitz prescription, which was originally proposed for general relativity. For such a choice of reference term, the referenced quasilocal m…
The $1/r$-expansion in the distance to the source is applied to the linearized $f(R)$ gravity, and its multipole expansion in the radiation field with irreducible Cartesian tensors is presented. Then, the energy, momentum, and angular momentum in the gravitational waves are provided for linearized $f(R)$ gravity. All of these results have two parts which are associated with the tensor part and the scalar part in the multipole expansion of linearized $f(R)$ gravity, respectively. The former is the same as that in General Relativity, and the latter, as the correction to the result in General Rel…
In general relativity, the local gravitational energy is best characterised by the quasilocal mass. The small sphere limit of quasilocal mass provides us the most local notion of gravitational energy. In four dimensions, the limits were shown be the stress tensor in non-vacuum and the Bel-Robinson tensor in vacuum. We study the local gravitational energy in higher dimensions through the lens of the small sphere limits of various quasilocal mass proposals which can be appropriately generalised beyond four dimensions, and report a new quantity $\Q$ which potentially characterises the local gravi…
The field equations of $f(R,\mathcal{G})$ gravity are rewritten in the form of obvious wave equations with the stress-energy pseudotensor of the matter fields and the gravitational field, as their sources, under the de Donder condition. The linearized field equations of $f(R,\mathcal{G})$ gravity are the same as those of linearized $f(R)$ gravity, and thus, their multipole expansions under the de Donder condition are also the same. It is also shown that the Gauss-Bonnet curvature scalar $\mathcal{G}$ does not contribute to the effective stress-energy tensor of gravitational waves in linearized…
The formalism for describing a metric and the corresponding scalar in terms of multipole moments has recently been developed for scalar-tensor theories. We take advantage of this formalism in order to obtain expressions for the observables that characterise geodesics in terms of the moments. These expressions provide some insight into how the structure of a scalarized compact object affects observables. They can also be used to understand how deviations from general relativity are imprinted on the observables.
This paper constructs the multipole expansion (in general relativity) of the gravitational field generated by a slowly-moving isolated source. We introduce some definitions for the source multipole moments, valid to all orders in a post-Newtonian expansion, and depending in a well-defined way on the total stress-energy pseudo-tensor of the material and gravitational fields. Previously obtained expressions of the source multipole moments are recovered in the appropriate limits. The source moments parametrize the linearized approximation of the gravitational field exterior to the source, as comp…
Stationary, asymptotically flat spacetimes in general relativity can be characterized by their multipole moments. The moments have proved to be very useful tools for extracting information about the spacetime from various observables and, more recently, for establishing universalities in the structure of neutron stars. As a first step toward extending these methods beyond general relativity, we develop the formalism that allows one to define and calculate the multipole moments in scalar-tensor theories of gravity.
This paper studies a class of $D=n+2(\ge 6)$ dimensional solutions to Lovelock gravity that is described by the warped product of a two-dimensional Lorentzian metric and an $n$-dimensional Einstein space. Assuming that the angular part of the stress-energy tensor is proportional to the Einstein metric, it turns out that the Weyl curvature of an Einstein space must obey two kinds of algebraic conditions. We present some exact solutions satisfying these conditions. We further define the quasilocal mass corresponding to the Misner-Sharp mass in general relativity. It is found that the quasilocal …
Angular Momentum Flux in Scalar-Tensor Theories up to 1.5 post-Newtonian Order
该研究利用多极矩计算了无质量标量-张量理论中非圆无自旋致密双星最高至1.5PN阶的角动量通量，分解了通量的构成并推导了尾效应与非线性记忆效应的表达式，为偏心双星的辐射反作用力和波形模板构建提供了基础。
An idealized "test" object in general relativity moves along a geodesic. However, if the object has a finite mass, this will create additional curvature in the spacetime, causing it to deviate from geodesic motion. If the mass is nonetheless sufficiently small, such an effect is usually treated perturbatively and is known as the gravitational self-force due to the object. This issue is still an open problem in gravitational physics today, motivated not only by basic foundational interest, but also by the need for its direct application in gravitational-wave astronomy. In particular, the observ…
The Multipolar Post-Minkowskian (MPM) formalism represents an approach for determining the metric density in the exterior of a compact source of matter. In the MPM formalism the metric density is given in harmonic coordinates and in terms of symmetric tracefree (STF) multipoles. In this investigation, the post-linear metric density of this formalism is used in order to determine the post-linear metric tensor in the exterior of a compact source of matter. The metric tensor is given in harmonic coordinates and in terms of STF multipoles. The post-linear metric coefficients are associated with an…
This paper employs the post-Newtonian approximations of scalar-tensor theory of gravity along with the Cartesian STF tensors and the Blanchet-Damour multipole formalism to derive translational and rotational equations of motion of N extended bodies with arbitrary distribution of mass and velocity. We assume that spacetime can be covered by a global coordinate chart which is Minkowskian at infinity. We also introduce N local coordinate charts adapted to each body and covering a finite domain of space around the body. Gravitational field of each body is parametrized by an infinite set of the bod…
The stress-energy distributional multipole for both uncharged and charged dust
该研究构建了不带电与带电的分布型应力-能量多极子张量，推导了其散度性质，证明其可通过挤压正则尘埃张量得到，且采用无坐标方式完成计算。
Curvature-Enhanced Inertia in Curved Spacetimes: An ADM-Based Formalism with Multipole Connections
本文提出基于ADM形式化和多极连接的曲率增强惯性张量定义，通过曲率修正和后牛顿贡献揭示了相对论中惯性与多极结构的统一几何描述。
The tensorial representation of the distributional stress-energy quadrupole and its dynamics
本文研究了由世界线上δ函数协变导数构造的应力-能量张量，推导了至四极阶的动力学方程，并证明了多极子分量的唯一性及其与张量场标准矩的关系。
Source multipoles and energy-momentum tensors for spinning black holes and other compact objects in arbitrary dimensions
本文在动量空间中推导出任意维度下由质量和角动量构成的能量-动量张量的一般形式，首次在广义相对论中定义源多极矩，并由此重建Kerr和Myers-Perry黑洞的多极结构及相应物质源。
Dynamical Implications of the Kerr Multipole Moments for Spinning Black Holes
本文证明Kerr解确定Dixon意义下旋转黑洞的全部多极矩及所有R线性耦合，并推导一般旋转物体至自旋高阶的电磁与引力Compton振幅。
In $f(R)$ gravity, the metric, presented in the form of the multipole expansion, for the external gravitational field of a spatially compact supported source up to $1/c^3$ order is provided, where $c$ is the velocity of light in vacuum. The metric consists of General Relativity-like part and $f(R)$ part, where the latter is the correction to the former in $f(R)$ gravity. At the leading pole order, the metric can reduce to that for a point-like or ball-like source. For the gyroscope moving around the source without experiencing any torque, the multipole expansions of its spin's angular velociti…
Multipole expansion of the gravitational field in a general class of fourth-order theories of gravity and the application in gyroscopic precession
本文在一般四阶引力理论中建立弱场慢运动近似，给出源外度规的多极展开及源多极矩闭式表达，并推导陀螺三类进动角速度的修正，发现该理论对 Lense-Thirring 进动给出非零修正，从而区别于 f(R) 类引力。
Multipole expansion at the level of the action in $d$-dimensions
本文在 (d+1) 维时空中研究辐射源长波有效作用量的多极展开，推导了标量、电磁和引力波的多极矩，包括奇宇称磁型矩和新的 Weyl 型矩，并在 d→3 极限下与已知结果一致，为引力波通量的有效场论计算奠定基础。
Entropy and energy are found to be closely tied on our quest for quantum gravity. We point out an interesting connection between the recently proposed outer entropy, a coarse-grained entropy defined for a compact spacetime domain motivated by the holographic duality, and the Bartnik-Bray quasilocal mass long known in the mathematics community. In both scenarios, one seeks an optimal spacetime fill-in of a given closed, connected, spacelike, codimension-two boundary. We show that for an outer-minimizing mean-convex surface, the Bartnik-Bray inner mass matches exactly with the irreducible mass c…
Gravitational Multipoles in Five Dimensions
本文将Thorne的ACMC坐标形式推广到五维，严格定义了稳态渐近平坦真空度规的质量与流多极矩，并在双轴对称下约化为$M_\ell$与$S^{(1)}_\ell, S^{(2)}_\ell$，成功应用于Myers-Perry黑洞、黑环及多中心几何。
This paper investigates the propagation of cosmological gravitational waves interacting with free-streaming neutrinos within the context of Horndeski theories of gravity constrained by the detection of GW170817. We apply the theory of cosmological perturbations to explicitly derive the Einstein-Boltzmann equation for the damped propagation of first-order transverse traceless gravitational waves. In contrast to general relativity, we argue that modified gravity can give rise to non-vanishing free-streaming damping effects during the cosmological matter dominated era. We also provide an analytic…
In recent years there has been a surge of interest in what has come to be known as the `universal relations' between various global properties of neutron stars. These universal relations are equation of state independent relations between quantities such as the moment of inertia $I$, the tidal deformability or Love number $λ$, and the quadrupole $Q$ (I-Love-Q relations), or the relativistic multipole moments (3-hair relations). While I-Love-Q relations have been studied extensively in both general relativity and various alternatives, 3-hair relations have been studied only in general relativit…
The problem of maintaining scale and conformal invariance in Maxwell and general N-form gauge theories away from their critical dimension d=2(N+1) is analyzed.We first exhibit the underlying group-theoretical clash between locality,gauge,Lorentz and conformal invariance require- ments. "Improved" traceless stress tensors are then constructed;each violates one of the above criteria.However,when d=N+2,there is a duality equivalence between N-form models and massless scalars.Here we show that conformal invariance is not lost,by constructing a quasilocal gauge invariant improved stress tensor.The …
We study the dynamical evolution of a scalar field coupling to Einstein's tensor in the background of Reissner-Nordström black hole. Our results show that the the coupling constant $η$ imprints in the wave dynamics of a scalar perturbation. In the weak coupling, we find that with the increase of the coupling constant $η$ the real parts of the fundamental quasinormal frequencies decrease and the absolute values of imaginary parts increase for fixed charge $q$ and multipole number $l$. In the strong coupling, we find that for $l\neq0$ the instability occurs when $η$ is larger than a certain thre…
$ϕ$-modes of neutron stars in a massless scalar-tensor theory
本文研究 Brans-Dicke 型无质量标量-张量理论中中子星的 $l=0$、1、2 标量主导 $ϕ$ 模，基于六种状态方程与广义相对论情形比较，并提取普适关系。
Sources of long wavelength radiation are naturally described by an effective field theory (EFT) which takes the form of a multipole expansion. Its action is given by a derivative expansion where higher order terms are suppressed by powers of the ratio of the size of the source over the wavelength. In order to determine the Wilson coefficients of the EFT, i.e. the multipole moments, one needs the mapping between a linear source term action and the multipole expansion form of the action of the EFT. In this paper we perform the multipole expansion to all orders by Taylor expanding the field in th…
We consider general perturbations of a Schwarzschild black holes in the context of f(R) gravity. A reduced set of frame independent master variables are determined, which obey two closed wave equations - one for the transverse, trace-free (tensor) perturbations and the other for the additional scalar degree of freedom which characterise fourth-order theories of gravity. We show that for the tensor modes, the underlying dynamics in f(R) gravity is governed by a modified Regge-Wheeler tensor which obeys the same Regge-Wheeler equation as in General Relativity. We find that the possible sources o…
The present paper derives the post-Newtonian Lagrangian of translational motion of N arbitrary-structured bodies with all mass and spin multipoles in a scalar-tensor theory of gravity. The multipoles depend on time and evolve in accordance with their own dynamic equations of motion. The Lagrangian is retrieved from the post-Newtonian equations of motion by solving the inverse problem of the Lagrangian mechanics and generalizes a well-known Lagrangian of pole-dipole-quadrupole massive particles to the particles of higher multipolarity. Analytic treatment of the higher-order multipole contributi…
Octupolar Gravitational Radiation in de Sitter Spacetime
该研究在广义谐和规范下推导了德西特时空局域物质源的八极引力辐射，提出通用多极矩微分恒等式，扩展了多极展开阶数，为构建多极后德西特形式奠定基础。
We consider the inflationary universe with a spectator scalar field coupled to a $U(1)$ gauge field and calculate curvature perturbation and gravitational waves (GWs). We find that the sourced GWs can be larger than the one from vacuum fluctuation and they are statistically anisotropic as well as linearly polarized. The GW power spectrum acquires higher multipole moments as $\mathcal{P}_h \propto (1-\cos^2θ+\cos^4θ-\cos^6θ)$ irrespective of the model parameters.
Probing gravity beyond general relativity with bispectrum multipoles of cosmological tracers: I. Theoretical Foundations
本文研究了红移空间双谱的非零角多极 $B_l^m$ 探测 DHOST 修改引力理论的潜力，发现 $l=2,4,6$ 多极比单极矩更敏感，且其符号反转可作为偏离广义相对论的稳健指标。
Scalar-Tensor Gravity as a Probe of Generalized Black Hole Entropy
本文通过将广义黑洞熵函数与Misner-Sharp准局部质量及Wald Noether-电荷熵建立直接联系，发展出一种几何实现，揭示了标量-张量引力中熵参数依赖的有效引力耦合，并推导出不同熵形式对应的标量势，为宇宙学现象提供新的理论框架。
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